be thought of as mountains, valleys, plains, and often pits. Possible solutions are found in the pits as they have the lower merit function values. With conventional optimization routines, the optical design program simply attempts to find the bottom of the pit local to the current location. However, the bottom of this pit may well not be the lowest and, consequently, not the optimum solution. (By optimum, we mean that the optical configuration solution having the smallest merit function value existing anywhere in the hyperspace; in other words, the global solution.)

Many of the optical design programs today include some form of what can generically be called global optimization. The objective of each searching approach these programs use is to locate the optimum solution or to give the designer a variety of potential solutions to consider. At times, “new” configurations have been found by allowing the number of elements and materials to vary. The achromat study just presented showed that there are exactly four perfect solutions for the merit function defined. A simple test that can be given to an optical design program is to find these four solutions. At least one optical design program is known to be able to automatically find these solutions.

7.3CORRECTION OF ZONAL SPHERICAL ABERRATION

If the zonal aberration in a lens system is found to be excessive, it can often be reduced by splitting the system into two lenses, each having half the lens power, in a manner analogous to the reduction of the marginal aberration of a single lens (see Section 6.1.6).

Another method that is frequently employed in a cemented system is to separate the cemented interface by a narrow parallel airgap. For this procedure to

be effective, there must be a large amount of spherical aberration in the airgap so that the marginal ray drops disproportionately rapidly as compared to the 0.7 zonal ray. The airgap therefore undercorrects the marginal aberration more rapidly than the zonal aberration. As the rear negative element is now not acting as strongly as before because of the reduction of incidence height, the last radius must be adjusted to restore the chromatic correction, ordinarily by use of the D d method. As the spherical aberration will now be strongly undercorrected, it must be restored by a bending of the whole lens. Using this procedure, it is often possible to correct both the marginal and the zonal aberrations simultaneously.

To determine the proper values of the airgap and the lens bending, we start with a cemented lens and introduce an arbitrary small parallel airgap, the last radius being found by the D – d method. The whole lens is then bent by trial until the marginal aberration is correct and the zonal aberration is found. If it is still negative, a wider airgap is required. The desired values are quickly found by plotting suitable graphs.

As an example, we may consider the following three f/3.3 systems. They each have a focal length of 10.0, and they are made from K-3 and F-4 glasses, the last radius in each case being found by the usual D – d procedure, as shown in Table 7.7.

System A is a well-corrected doublet of the ordinary type, but of unusually high aperture so as to illustrate the principle. The spherical aberration curve is shown as A in Figure 7.5. After introducing an airgap and suitably strengthening the last radius by the D – d method, we have System B. The change in aberrations as a result of the introduction of this airgap is

adjusting the airgap to avoid the introduction of yet higher-order aberration terms. Somewhat improved performance can be achieved by shifting the zero zonal aberration point to a bit higher value of r. A problem likely to arise is that at least quintic aberration will now appear and have a rather significant value. The presence of the higher-order aberration makes the lens less tolerant to manufacturing and alignment errors.

When System A, after introducing a small airgap, was optimized by a typical lens design program using the same criteria as used in the preceding procedure, the resulting design was found to be quite similar, with the airgap being about one-third of System C. Figure 7.6 illustrates the longitudinal aberration and should be compared with curve C in Figure 7.5. It should be mentioned that there are many similar designs that have essentially the same performance as the airgap is varied and the curvatures are readjusted.

DESIGNER NOTE

An alternative procedure that can be applied to reduce zonal aberration is to thicken a lens element, provided there is a large amount of undercorrected aberration within the glass. This is done frequently in photographic objectives, such as in Double-Gauss lenses of high aperture. Of course, introducing an air space by breaking cemented surfaces can be done in concert with element thickening.

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Figure 7.6 Longitudinal spherical aberration for an achromatic doublet having a small airgap designed using a computer optical design program.